\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  -2.0 & -6.00 x_{1} & +  2.00 x_{2} & +  3.00 x_{3} & -2.00 x_{4} & + 10.00 x_{5} & -9.00 x_{6} & +  2.00 x_{7} & +  4.00 x_{8} &   \\
 x_{11}   &  0.0 & +  7.00 x_{1} & +  1.00 x_{2} & + 10.00 x_{3} & +  9.00 x_{4} & -3.00 x_{5} & -9.00 x_{6} & +  5.00 x_{7} & -10.00 x_{8} & -5.00 x_{9}\\
 x_{12}   &  2.0 & +  3.00 x_{1} & -10.00 x_{2} &   & -3.00 x_{4} & + 10.00 x_{5} & -3.00 x_{6} & +  9.00 x_{7} & +  3.00 x_{8} & -4.00 x_{9}\\
 x_{13}   &  3.0 & +  9.00 x_{1} & +  4.00 x_{2} & -2.00 x_{3} & +  8.00 x_{4} & -2.00 x_{5} & -3.00 x_{6} &   & -7.00 x_{8} & +  5.00 x_{9}\\
 x_{14}   &  3.0 & +  2.00 x_{1} & +  7.00 x_{2} & +  6.00 x_{3} & -7.00 x_{4} & -4.00 x_{5} & +  7.00 x_{6} & -5.00 x_{7} & +  1.00 x_{8} & +  2.00 x_{9}\\
\hline
z    &  0.0 & -4.00 x_{1} & -4.00 x_{2} & +  4.00 x_{3} & -1.00 x_{4} & -1.00 x_{5} &   & +  4.00 x_{7} & +  2.00 x_{8} & +  5.00 x_{9}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{9}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,14$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  6.00 y_{10} & -7.00 y_{11} & -3.00 y_{12} & -9.00 y_{13} & -2.00 y_{14}\\
 y_{2}   &  1.0 & -2.00 y_{10} & -1.00 y_{11} & + 10.00 y_{12} & -4.00 y_{13} & -7.00 y_{14}\\
 y_{3}   &  1.0 & -3.00 y_{10} & -10.00 y_{11} &   & +  2.00 y_{13} & -6.00 y_{14}\\
 y_{4}   &  1.0 & +  2.00 y_{10} & -9.00 y_{11} & +  3.00 y_{12} & -8.00 y_{13} & +  7.00 y_{14}\\
 y_{5}   &  1.0 & -10.00 y_{10} & +  3.00 y_{11} & -10.00 y_{12} & +  2.00 y_{13} & +  4.00 y_{14}\\
 y_{6}   &  1.0 & +  9.00 y_{10} & +  9.00 y_{11} & +  3.00 y_{12} & +  3.00 y_{13} & -7.00 y_{14}\\
 y_{7}   &  1.0 & -2.00 y_{10} & -5.00 y_{11} & -9.00 y_{12} &   & +  5.00 y_{14}\\
 y_{8}   &  1.0 & -4.00 y_{10} & + 10.00 y_{11} & -3.00 y_{12} & +  7.00 y_{13} & -1.00 y_{14}\\
 y_{9}   &  1.0  &   & +  5.00 y_{11} & +  4.00 y_{12} & -5.00 y_{13} & -2.00 y_{14}\\
\hline
z    &  -0 & +  2.00 y_{10} &   & -2.00 y_{12} & -3.00 y_{13} & -3.00 y_{14}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 3 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.26605504587 & -0.74 y_{5} & +  0.48 y_{3} & -10.43 y_{12} & -8.47 y_{13} & +  3.83 y_{14}\\
 y_{2}   &  0.697247706422 & +  0.16 y_{5} & +  0.15 y_{3} & + 11.56 y_{12} & -4.61 y_{13} & -6.74 y_{14}\\
 y_{11}   &  0.0642201834862 & +  0.03 y_{5} & -0.09 y_{3} & +  0.28 y_{12} & +  0.13 y_{13} & -0.66 y_{14}\\
 y_{4}   &  0.660550458716 & -0.43 y_{5} & +  0.77 y_{3} & -1.31 y_{12} & -8.68 y_{13} & + 13.35 y_{14}\\
 y_{10}   &  0.119266055046 & -0.09 y_{5} & -0.03 y_{3} & -0.92 y_{12} & +  0.24 y_{13} & +  0.20 y_{14}\\
 y_{6}   &  2.65137614679 & -0.58 y_{5} & -1.07 y_{3} & -2.78 y_{12} & +  6.30 y_{13} & -11.13 y_{14}\\
 y_{7}   &  0.440366972477 & +  0.05 y_{5} & +  0.51 y_{3} & -8.54 y_{12} & -1.12 y_{13} & +  7.90 y_{14}\\
 y_{8}   &  1.16513761468 & +  0.64 y_{5} & -0.81 y_{3} & +  3.42 y_{12} & +  7.33 y_{13} & -8.41 y_{14}\\
 y_{9}   &  1.32110091743 & +  0.14 y_{5} & -0.46 y_{3} & +  5.38 y_{12} & -4.36 y_{13} & -5.30 y_{14}\\
\hline
z    &  0.238532110092 & -0.18 y_{5} & -0.06 y_{3} & -3.83 y_{12} & -2.52 y_{13} & -2.60 y_{14}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.183486238532 & +  0.74 x_{1} & -0.16 x_{2} & -0.03 x_{11} & +  0.43 x_{4} & +  0.09 x_{10} & +  0.58 x_{6} & -0.05 x_{7} & -0.64 x_{8} & -0.14 x_{9}\\
 x_{3}   &  0.0550458715596 & -0.48 x_{1} & -0.15 x_{2} & +  0.09 x_{11} & -0.77 x_{4} & +  0.03 x_{10} & +  1.07 x_{6} & -0.51 x_{7} & +  0.81 x_{8} & +  0.46 x_{9}\\
 x_{12}   &  3.83486238532 & + 10.43 x_{1} & -11.56 x_{2} & -0.28 x_{11} & +  1.31 x_{4} & +  0.92 x_{10} & +  2.78 x_{6} & +  8.54 x_{7} & -3.42 x_{8} & -5.38 x_{9}\\
 x_{13}   &  2.52293577982 & +  8.47 x_{1} & +  4.61 x_{2} & -0.13 x_{11} & +  8.68 x_{4} & -0.24 x_{10} & -6.30 x_{6} & +  1.12 x_{7} & -7.33 x_{8} & +  4.36 x_{9}\\
 x_{14}   &  2.59633027523 & -3.83 x_{1} & +  6.74 x_{2} & +  0.66 x_{11} & -13.35 x_{4} & -0.20 x_{10} & + 11.13 x_{6} & -7.90 x_{7} & +  8.41 x_{8} & +  5.30 x_{9}\\
\hline
z    &  -0.238532110092 & -1.27 x_{1} & -0.70 x_{2} & -0.06 x_{11} & -0.66 x_{4} & -0.12 x_{10} & -2.65 x_{6} & -0.44 x_{7} & -1.17 x_{8} & -1.32 x_{9}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.183486238532 & +  0.74 x_{1} & -0.16 x_{2} & -0.03 x_{11} & +  0.43 x_{4} & +  0.09 x_{10} & +  0.58 x_{6} & -0.05 x_{7} & -0.64 x_{8} & -0.14 x_{9}\\
 x_{3}   &  0.0550458715596 & -0.48 x_{1} & -0.15 x_{2} & +  0.09 x_{11} & -0.77 x_{4} & +  0.03 x_{10} & +  1.07 x_{6} & -0.51 x_{7} & +  0.81 x_{8} & +  0.46 x_{9}\\
 x_{12}   &  3.83486238532 & + 10.43 x_{1} & -11.56 x_{2} & -0.28 x_{11} & +  1.31 x_{4} & +  0.92 x_{10} & +  2.78 x_{6} & +  8.54 x_{7} & -3.42 x_{8} & -5.38 x_{9}\\
 x_{13}   &  2.52293577982 & +  8.47 x_{1} & +  4.61 x_{2} & -0.13 x_{11} & +  8.68 x_{4} & -0.24 x_{10} & -6.30 x_{6} & +  1.12 x_{7} & -7.33 x_{8} & +  4.36 x_{9}\\
 x_{14}   &  2.59633027523 & -3.83 x_{1} & +  6.74 x_{2} & +  0.66 x_{11} & -13.35 x_{4} & -0.20 x_{10} & + 11.13 x_{6} & -7.90 x_{7} & +  8.41 x_{8} & +  5.30 x_{9}\\
\hline
z    &  0.0366972477064 & -6.65 x_{1} & -4.43 x_{2} & +  0.39 x_{11} & -4.51 x_{4} & +  0.02 x_{10} & +  3.72 x_{6} & +  1.99 x_{7} & +  5.87 x_{8} & +  6.97 x_{9}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.183486238532 & +  0.74 x_{1} & -0.16 x_{2} & -0.03 x_{11} & +  0.43 x_{4} & +  0.09 x_{10} & +  0.58 x_{6} & -0.05 x_{7} & -0.64 x_{8} & -0.14 x_{9}\\
 x_{3}   &  0.0550458715596 & -0.48 x_{1} & -0.15 x_{2} & +  0.09 x_{11} & -0.77 x_{4} & +  0.03 x_{10} & +  1.07 x_{6} & -0.51 x_{7} & +  0.81 x_{8} & +  0.46 x_{9}\\
 x_{12}   &  3.83486238532 & + 10.43 x_{1} & -11.56 x_{2} & -0.28 x_{11} & +  1.31 x_{4} & +  0.92 x_{10} & +  2.78 x_{6} & +  8.54 x_{7} & -3.42 x_{8} & -5.38 x_{9}\\
 x_{13}   &  2.52293577982 & +  8.47 x_{1} & +  4.61 x_{2} & -0.13 x_{11} & +  8.68 x_{4} & -0.24 x_{10} & -6.30 x_{6} & +  1.12 x_{7} & -7.33 x_{8} & +  4.36 x_{9}\\
 x_{14}   &  2.59633027523 & -3.83 x_{1} & +  6.74 x_{2} & +  0.66 x_{11} & -13.35 x_{4} & -0.20 x_{10} & + 11.13 x_{6} & -7.90 x_{7} & +  8.41 x_{8} & +  5.30 x_{9}\\
\hline
z    &  0.0366972477064 & -6.65 x_{1} & -4.43 x_{2} & +  0.39 x_{11} & -4.51 x_{4} & +  0.02 x_{10} & +  3.72 x_{6} & +  1.99 x_{7} & +  5.87 x_{8} & +  6.97 x_{9}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.414847161572 & +  1.52 x_{1} & +  0.27 x_{2} & -0.04 x_{11} & +  1.23 x_{4} & +  0.07 x_{10} & -0.09 x_{13} & +  0.06 x_{7} & -1.31 x_{8} & +  0.26 x_{9}\\
 x_{3}   &  0.484716157205 & +  0.97 x_{1} & +  0.64 x_{2} & +  0.07 x_{11} & +  0.71 x_{4} & -0.01 x_{10} & -0.17 x_{13} & -0.32 x_{7} & -0.44 x_{8} & +  1.20 x_{9}\\
 x_{12}   &  4.94759825328 & + 14.17 x_{1} & -9.53 x_{2} & -0.33 x_{11} & +  5.14 x_{4} & +  0.81 x_{10} & -0.44 x_{13} & +  9.03 x_{7} & -6.66 x_{8} & -3.45 x_{9}\\
 x_{6}   &  0.400291120815 & +  1.34 x_{1} & +  0.73 x_{2} & -0.02 x_{11} & +  1.38 x_{4} & -0.04 x_{10} & -0.16 x_{13} & +  0.18 x_{7} & -1.16 x_{8} & +  0.69 x_{9}\\
 x_{14}   &  7.05094614265 & + 11.12 x_{1} & + 14.87 x_{2} & +  0.43 x_{11} & +  1.98 x_{4} & -0.62 x_{10} & -1.77 x_{13} & -5.92 x_{7} & -4.53 x_{8} & + 13.00 x_{9}\\
\hline
z    &  1.52401746725 & -1.66 x_{1} & -1.72 x_{2} & +  0.32 x_{11} & +  0.60 x_{4} & -0.12 x_{10} & -0.59 x_{13} & +  2.65 x_{7} & +  1.55 x_{8} & +  9.54 x_{9}\\
\end{array}\]


 $ x_{4} $ enters and Unbounded Dictionary!


 $ x_{4} $ enters and Unbounded Dictionary!
\end{document}
